Some Modifications of Pairwise Soft Sets and Some of Their Related Concepts

Samer Al Ghour

doi:10.3390/math9151781

Abstract

In this paper, we first define soft u-open sets and soft s-open as two new classes of soft sets on soft bitopological spaces. We show that the class of soft p-open sets lies strictly between these classes, and we give several sufficient conditions for the equivalence between soft p-open sets and each of the soft u-open sets and soft s-open sets, respectively. In addition to these, we introduce the soft u-ω-open, soft p-ω-open, and soft s-ω-open sets as three new classes of soft sets in soft bitopological spaces, which contain soft u-open sets, soft p-open sets, and soft s-open sets, respectively. Via soft u-open sets, we define two notions of Lindelöfeness in SBTSs. We discuss the relationship between these two notions, and we characterize them via other types of soft sets. We define several types of soft local countability in soft bitopological spaces. We discuss relationships between them, and via some of them, we give two results related to the discrete soft topological space. According to our new concepts, the study deals with the correspondence between soft bitopological spaces and their generated bitopological spaces.

Highlights

Introduction and PreliminariesThis paper follows the notions and terminologies that appeared in [1,2]
Via soft p-open sets, several soft bitopological concepts were introduced by various researchers
We first define soft u-open sets and soft s-open as two new classes of soft sets on soft bitopological spaces, and we show that the class of soft p-open sets lies strictly between these classes

Summary

Introduction and Preliminaries

This paper follows the notions and terminologies that appeared in [1,2]. In this paper, TS SS(U, E) denotes the family of all soft sets over U relative to E. Denote the family of all ω-open sets in the TS (U, =) by =ω. Three types of ω-open sets were defined and studied in bitopological spaces in [30]. Soft ω-open sets were defined and investigated in STSs in [2], and research via them was continued in [5,31]. The concept of soft pairwise open (soft p-open) sets in SBTSs was defined and studied. Via soft p-open sets, several soft bitopological concepts were introduced by various researchers.

Soft u-Open Sets and Soft s-Open Sets in SBTSs

Soft Local Countability

Conclusions